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      Very Large-Scale Singular Value Decomposition Using Tensor Train Networks

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          Abstract

          We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant singular values and corresponding singular vectors for large-scale structured matrices given in a TT format. The computational complexity of the proposed methods scales logarithmically with the matrix size under the assumption that both the matrix and the singular vectors admit low-rank TT decompositions. The proposed methods, which are called the alternating least squares for SVD (ALS-SVD) and modified alternating least squares for SVD (MALS-SVD), compute the left and right singular vectors approximately through block TT decompositions. The very large-scale optimization problem is reduced to sequential small-scale optimization problems, and each core tensor of the block TT decompositions can be updated by applying any standard optimization methods. The optimal ranks of the block TT decompositions are determined adaptively during iteration process, so that we can achieve high approximation accuracy. Extensive numerical simulations are conducted for several types of TT-structured matrices such as Hilbert matrix, Toeplitz matrix, random matrix with prescribed singular values, and tridiagonal matrix. The simulation results demonstrate the effectiveness of the proposed methods compared with standard SVD algorithms and TT-based algorithms developed for symmetric eigenvalue decomposition.

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          Author and article information

          Journal
          2014-10-25
          2014-12-13
          Article
          10.1137/140983410
          1410.6895
          f5a4a19f-f956-4937-8df9-907c6d45ff92

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          15A18, 65F15, 65F30
          SIAM. J. Matrix Anal. Appl. 36(3):994-1014, 2015
          math.NA cs.NA

          Numerical & Computational mathematics
          Numerical & Computational mathematics

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